RUI: Hurwitz Number Fields

数论和几何是数学中最古老的两个领域。数域在数论中起着核心作用，而曲面在几何学中起着核心作用。该研究项目涉及从曲面几何自然产生的数字领域，其构建可以追溯到数学家阿道夫·赫维茨在1900年左右的工作。这些Hurwitz数域具有从纯粹数论的观点来看非常不寻常的性质的组合：它们的次数和伽罗瓦群通常非常大，而它们的分枝素数集非常小。事实上，关于这些不变量，赫维茨数域是如此极端，以至于它们的存在与最近成功指导其他数域研究的启发式相矛盾。这项研究项目的目标是建立这些外围数字字段都以统一和高度结构化的方式运行。该项目的结果预计将支持围绕数论的几个学科相互关联的未来发展。该项目包括与三名本科生的合作，专门为他们的科学职业生涯做好准备。这些合作将作为学生未来参与高级数学研究的模型。赫维茨数域作为黎曼曲面到黎曼球面的映射的定义域而产生。该研究项目的主要焦点将集中在任意Hurwitz数域上分支的显式描述上。从定义场的几何数据中，人们可以读出有限的素数集，在这些素数集之外，已知分支是驯服的。该项目将寻求在这些驯服的素数上的分支的精确公式和在剩余的素数上的野生分支的尖锐上界。除了研究分支，研究人员还将致力于建立一个关于给定的分支素数集的度的无界性的猜想的几何类比。在不同的方向上，允许分支素数集增长，调查者的目标是通过找到同时定义无限多个Hurwitz数域的多元多项式的多指标序列来进一步证明行为的一致性。

Two of the oldest areas of mathematics are number theory and geometry. Number fields play a central role in number theory, while surfaces play a central role in geometry. The research project concerns number fields that arise naturally from the geometry of surfaces, via a construction dating back to work of the mathematician Adolph Hurwitz around 1900. These Hurwitz number fields have a combination of properties that is very unusual from a purely number-theoretic point of view: their degrees and Galois groups are typically very large, while their sets of ramifying primes are very small. In fact, Hurwitz number fields are so extreme with respect to these invariants that their existence contradicts a heuristic that has successfully guided other recent research into number fields. The goal of this research project is to establish that these outlying number fields all behave in a uniform and highly structured way. The results of the project are expected to support future developments interrelating several disciplines surrounding number theory. The project includes collaborations with three undergraduates, designed specifically to better prepare them for scientific careers. These collaborations will serve as models for future involvement of students in high-level mathematics research.Hurwitz number fields arise as fields of definition of maps for Riemann surfaces to the Riemann sphere. The main focus of the research project will be on the explicit description of ramification in an arbitrary Hurwitz number field. From the geometric data defining the field, one can read off a finite set of primes outside of which ramification is known to be tame. The project will pursue exact formulas for ramification at these tame primes and sharp upper bounds for the wild ramification at the remaining primes. Besides studying ramification, the investigator will aim to establish a geometric analog of a conjecture on the unboundedness of degrees for a given set of ramifying primes. In a different direction, where the set of ramifying primes is allowed to grow, the investigator aims to demonstrate further uniformity of behavior by finding multi-indexed sequences of multivariate polynomials that define infinitely many Hurwitz number fields at once.